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Full text of "Treatise On Analysis Vol-Ii"

4    UNIFORMIZABLE SPACES       11

family of pseudo-distances (dx\ e , . A topology which can be defined by a family
of pseudo-distances is said to be uniformizable, and a space equipped with such
a topology is said to be uniformizable. It is obvious that a metrizable space is
uniformizable, but the converse is false (for example, the family consisting
only of the pseudo-distance J = 0 defines the chaotic topology). There exist
Hausdorff topologies which are not uniformizable (Problems 3 and 4); but
throughout this book (and in the majority of situations in analysis) uniformiz-
able spaces are the only ones which we shall have to consider.

(12.4.2)   Let E be a uniformizable space whose topology is defined by a family
of pseudo-distances (da)ael . Then the sets

B(a; (*j\ (/v))       (resp. B'<*; (ay), (r,)))
are open (resp. closed) in E.

Let x e B(a; (a,-), (r,.)), and put Sj = daj(a, x). Then Sj < r,- for 1 <>j  m,
hence B(a; (a,-), (/y)) contains the set E(x; (a/), (r^  Sj)) by virtue of the tri-
angle inequality for each of the daj . This shows that E(a; (ocj), (rj)) is an open
set. If x $ B'(#; (o/) (r,-)), then there exists an index k such that 1 g k <J m and
dak(a, x) = sk > rk , and hence the triangle inequality for dak shows that the set
B(x\ ak,sk~~ rk) does not meet B'(<^; (o/X (o))* This shows that Br((x;(aj), (r^))
is closed.

(12.4.3)    In a uniformizable space, the closed neighborhoods of a point form a
fundamental system of neighborhoods of the point,

This follows from (12.4.2) and the fact that

B'O; (a,), (0/2)) <= E(a; (a,), (ry)).

(12.4.4) ^r E be a uniformizable space and let (rfa)ae j 6# a family of pseudo-
distances defining the topology ofE. Then E is Hausdorff if and only if, for each
pair of distinct points x, y in Ey there exists ft e I such that dfl(x, y) / 0.

If E is Hausdorff, then by definition there exists a set B(JC; (ay), (ry)) not
containing y, and therefore for at least one index/ we have daj(x, y)^rj> 0.
Conversely, itd^x, y) = / > 0, then the open neighborhoods B(x; J8, r/2) and
B(j; j^? ^/2) of x and 7, respectively, are disjoint, by virtue of the triangle
inequality for dp .

If (4) is a family of pseudo-distances defining a Hausdorff topology on a
set E, then to say that a sequence (xn) of points of E has a point a as limit with
respect to this topology means that lim da(a, xn) = Ofor all a.section of the